EM.Tempo Tutorial Lesson 6: Modeling Coplanar Waveguide Transmission Lines and Active Devices

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Tutorial Project: Modeling Circuits with Lumped Elements
FDTD205.png

Objective: In this project, coplanar waveguide (CPW) transmission lines with passive and active lumped loads and different waveforms are examined.

Concepts/Features:

  • Lumped Source
  • Lumped Load
  • Coupled Ports
  • Resistor
  • Diode
  • Field Probe
  • Waveform
  • S-Parameters
  • Voltage Standing Wave Ratio
  • Variables
  • Parametric Sweep

Minimum Version Required: All versions

'Download2x.png Download Link: [1]

What You Will Learn

In this tutorial you will learn how to construct coplanar waveguide (CPW) lines and use coupled port definitions. You will also introduce passive and active lumped sources like resistors and diodes into your circuits. You will use different waveforms to excite your structure and will observe the time domain signals using field probes.

Getting Started

Open the EM.Cube application and switch to FDTD Module. Start a new project with the following attributes:

Starting Parameters
Name FDTDLesson6
Length Units Millimeters
Frequency Units GHz
Center Frequency 3GHz
Bandwidth 3GHz

A coplanar waveguide with a slot width of 1mm and center metal strip of 2mm on a dielectric substrate of 1.2mm thick and εr = 2.2 has a characteristic impedance of Z0 100Ω and effective permittivity of εeff 1.44. At an operating frequency of fo = 3GHz, free-space and guide wavelengths are λ0 = 100mm and λg = λ0/√εeff = 83.4mm , respectively. You can verify these results using the Transmission Line Calculator tool in the Device Manager of RF.Spice A/D, if you have installed it on your computer. You can access the Device Manager directly from the Tools Menu of EM.Tempo.

The Transmission Line Calculator tool of RF.Spice A/D.

Building a CPW Transmission Line

Build a thin dielectric substrate of material ROGER RT/Duroid 5880 (εr = 2.2) with a thickness of 1.2mm. Draw three parallel PEC rectangle strip objects and four PEC vertical line objects with the coordinates, rotation angles and dimensions given in the table below. The three PEC strips form your CPW transmission line. Rect_Strip_1 is the center conductor and Rect_Strip_2 and Rect_Strip_3 are the lateral ground planes. Line_1 and Line_2 are drawn in opposite directions and will be used to define the lumped sources. Line_3 and Line_4 are also drawn in opposite directions and will be used to define the termination lumped loads.

Part Object Type Material Type Dimensions Coordinates Rotation Angles
Box_1 Box ROGER RT/Duroid 5880 125mm × 84mm × 1.2mm (0, 0, 0) (0°, 0°, 0°)
Rect_Strip_1 Rectangle PEC 100mm × 2mm (0, 0, 1.2mm) (0°, 0°, 0°)
Rect_Strip_2 Rectangle PEC 100mm × 40mm (0, 22mm, 1.2mm) (0°, 0°, 0°)
Rect_Strip_3 Rectangle PEC 100mm × 40mm (0, -22mm, 1.2mm) (0°, 0°, 0°)
Line_1 Line PEC 1mm (-50mm, 1mm, 1.2mm) (0°, 0°, 90°)
Line_2 Line PEC 1mm (-50mm, -1mm, 1.2mm) (0°, 0°, -90°)
Line_3 Line PEC 1mm (50mm, 1mm, 1.2mm) (0°, 0°, 90°)
Line_4 Line PEC 1mm (50mm, -1mm, 1.2mm) (0°, 0°, -90°)
The geometry of the CPW line segment.

Defining the Lumped Sources, Lumped Loads and Coupled Ports

You have to excite the odd mode of your CPW transmission line. Define two lumped sources: LS_1 on Line_1 and LS_2 on Line_2, both with a Source Resistance of 200Ω. Note that since the two lines are oriented in the opposite directions, their lumped sources point in the +Y and -Y directions, respectively. In this way, this leads to the excitation of the odd mode of the CPW line.

The Lumped Source dialog for the first source.
The Lumped Source dialog for the second source.

Next, define two lumped loads of Resistor type: LL_1 on Line_3 and LL_2 on Line_3. To define a lumped load, right-click on the Lumped Loads section of the navigation tree and insert a new load from the contextual menu. In the Lumped Load dialog, select Resisitor from the Type dropdown list. Choose the right line object for each load. The default value of Resistance is 100Ω. Replace this value with the text string "RR" to define a new variable. This will introduce a new variable with an initial value of 100. Use "RR" as the value of both lumped loads. Although the line objects Line_3 and Line_4 have opposite directions, the polarity doesn't matter for resistive loads.

The Lumped Load dialog for the first resistor.
The Lumped Load dialog for the second resistor.
Attention icon.png In CPW structures, two source impedances are shunted at the source location. Therefore, for a CPW line with a characteristics impedance of Z0, you need to set each source resistance to 2Z0. Similarly, to define a resistive load of RL, you need to shunt two lumped loads with a resistance value of 2Z0 at the desired load location.

Next, initiate a Port Definition observable. In the Port Definition dialog, initially you will see two default ports: Port_1 associated with LS_1 and Port_2 associated with LS_2. For your CPW line, you are going to couple these two ports into a single port. Select Port_2 in Port Definition dialog and delete it. Then select Port_1 and click the Edit button. In the Edit Port dialog, you will see a list of "Available" sources in the table on the left, which contains "LS_2". Select it and move it to the "Associated" sources table on the right using the --> button. Also, change the reference Impedance of the port to 100Ω. Once you return to the Port Definition dialog, you will see that both LS_1 and LS_2 have now been associated with Port_1. In other words, you have defined a single coupled port called Port_1. This means that your structure has only one scattering parameter: S11.

The Port Definition dialog showing two coupled sources.
The Edit Port dialog.

Defining & Editing Variables

In EM.Cube, variables can be defined in two different ways. The first one is the formal way of defining a variable in the Variables Dialog and then assigning it to the object parameters. The second method is much simpler. Just open the property dialog of any object and simply type in a letter or string of letters to replace a numeric value. The string will be interpreted as a new variable and is automatically added to the variable list. For this project and for the resistor values, you used the second method in the previous section.

To verify your variable, click the Variables button of the Simulate Toolbar or use the keyboard shortcut Ctrl+B to open the Variables Dialog. Beside the global project variables fc (center frequency) and bw (bandwidth) and a few others, you will see your new variable "RR" in the variables list with it current value of 100. As we mentioned above, you need to set the resistor values to 200Ω if you want to have a matched termination load. Select "RR" in the list and click the Edit button. In the Edit Variable dialog, enter 200 in the Definition box. Once you return to the Variables dialog, you will see that the current value of RR has changed to 200.

Attention icon.png You can use a variable's name directly or any mathematical expression involving variable names to assign a new value for most object parameters.
The Variables dialog showing your resistance variable.
Change the definition of your variable in the Edit Variable dialog.

Running an FDTD Analysis

Before you run an FDTD simulation of your CPW structure, do the following:

  • Reduce the size of the computational domain by changing the default Offset values in all six directions to 0.1λ0 in the Domain Settings dialog.
  • Define three Field Sensor observables according to the following table:
Field Sensor Direction Coordinates
Sensor_1 Z (0, 1.5mm, 1.2mm)
Sensor_2 X (0, 1.5mm, 1.2mm)
Sensor_3 Y (0, 1.5mm, 1.2mm)
  • Open the Mesh Settings dialog, click the button labeled High Precision Mesh Settings, change the mesh density to 100 Cells/λeff, and change the value of Absolute Minimum Grid Spacing to 0.02 as a fraction of Max Grid Spacing in free space.
The drawn structure with its source, load and observables.

Open the FDTD Run Dialog and change the Power Threshold Level of the default value of -30dB to -50dB. This is usually a good choice with resonant structures that exhibit an oscillatory behavior. Run an FDTD simulation of your transmission line circuit and visualize its field distributions. The figure below shows the electric field distribution on a horizontal plane at z = 1.2mm, i.e. the surface of the CPW line. The E-field is almost zero everywhere except on the two slots. The fields are uniform longitudinally along the two slot lines, meaning that a decent impedance match has been accomplished and there is little wave reflection that would cause a standing wave pattern.

The intensity plot of the electric field distribution on a horizontal plane at Z = 1.2mm.

You can display field sensor data as vector plots. To do so, open the property dialog a the vertical field sensor Sensor_2 and select Vector for the Plot Type. Set the value of "Maxz Size" to 1 and value of "Cone Ratio" to 0.5.

Teh vector plot of the electric field distribution on a vertical plane at X = 0.

In the figure below, you can clearly see the field lines on and around the CPW slot lines.

Teh vector plot of the electric field distribution on a vertical plane at X = 0.

Next, open the Data Manager and plot the S11 and Z11 parameters in EM.Grid. As you can see from the figure below, the return loss |S11| is below -15dB, which implies a good impedance match.

The graph of S11 parameter of the CPW line segment.
The graph of Z11 parameter of the CPW line segment.

Changing the Termination Load

In this section of the tutorial lesson, you are going to change the value of the termination load RR and see how it affects the field distribution along the CPW as well as its port characteristics. Open the Variables dialog and change the value of RR to 1e+6, modeling an open end. Run a new FDTD simulation and visualize the field and plot the S11 and Z11 parameters. Open the property dialog of the E_total plot on the navigation tree and change the upper limit of the plot to 1e+3. There is a large E-field at the open end. You can see the standing wave pattern much better after lowering the upper limit.

The electric field distribution on a horizontal plane at Z = 1.2mm with an open end termination (RR = 1.e+6Ω).

You can also plot the 2D field profile graphs in EM.Grid. Open EM.Cube's Data Manager and plot the data file "Sensor_1_X_ETotal.DAT" as shown below. From the transmission line theory, you know that the distance between two consecutive peaks of the standing pattern is λg/2. You can measure this distance on the graph in EM.Grid using its "Delta Line Mode". This tool can measure the X and Y difference between any two points on any graph. Click the "Delta Line Mode" B2Delta Tool.png button of the Graph Toolbar. Place the mouse at a positive peak of the graph and click the left mouse button and then drag it to its next positive peak. The Delta X and Delta Y values are displayed in the Status Bar. You will read a Delta X value of 41.172mm. This value agrees very will RF.Spice's calculated value of λg = 83.4mm. Also, from the number of peaks (N = 2.5), you can say that the length of the transmission line must be about 1.25λg, which is also true.

2D Cartesian graph of electric field distribution along the X-axis on the horizontal plane at Z = 1.2mm with RL,eff → ∞.

The figure below shows the graph of return loss of your circuit. As you would have expected from an open-ended transmission line segment, the magnitude of the reflection coefficient is |S11| = 1.

The graph of S11 parameter of the open-ended CPW line segment.

Next, change the value of the variable RR to 50, meaning an effective load resistance of 25Ω. Run a new FDTD simulation and visualize the results. The figure below shows the new electric field distribution along the X-axis on the horizontal plane at Z = 1.2mm. Note that as opposed to the previous case of open ends, the field does not vanish completely at any point along the slots when the termination load has a finite non-zero resistance.

2D Cartesian graph of electric field distribution along the X-axis on the horizontal plane at Z = 1.2mm with RL,eff = 25Ω.

The figure below shows the return loss of the CPW line segment terminated at RL,eff = 25Ω.

From the transmission line theory, you know that the input reflection coefficient of your terminated line segment can be expressed as:

[math]\Gamma_{in} = \Gamma_L e^{-2j \beta L} = \frac{\zeta_L - 1} {\zeta_L + 1} e^{-2j \beta L} [/math]

where ΓL is the load reflection coefficient, ζL = ZL/Z0 is the normalized load value, and L is the length of the line segment. In this project, L = 100mm, ζL = 25Ω/100Ω = 0.25. Therefore, you will find | Γin | = | Γin | = 0.6 = -4.44dB.

The graph of S11 parameter of the CPW line segment terminated at RL,eff = 25Ω.

Exploring Time Domain Waveforms

By default, EM.Cube's FDTD Module uses a modulated Gaussian waveform for the excitation source. To examine the time domain response of your circuit, define a Field Probe as a new observable. Place it at (0, 0, 1mm) in the middle of the line half-way between the microstrip and ground. Set its direction to Z to probe the Ez and Hz fields at this location. Delete the three field sensor observables and the port definition. Run a new FDTD simulation. After the completion of the simulation, plot the data files "Probe_1_E_Time.DAT" and "Probe_1_H_Time.DAT" in EM.Grid as shown below:


Cartesian graph of the Ez field component as a function of time.


Cartesian graph of the Ez field component as a function of time.
Cartesian graph of the Ez field component as a function of time.


Cartesian graph of the Ez field component as a function of time.
Cartesian graph of the Ez field component as a function of time.


Cartesian graph of the Ez field component as a function of time.










Cartesian graph of the Ez field component as a function of time.
Cartesian graph of the Hz field component as a function of time.

The E-field plot shows that the incident signal reaches the probe location at about t = 1ns and it also shows its multiple reflections from the unmatched load. Note that the time between the incident signal peak and the first reflection is about 0.67ns. This is expected as the signal travels a total distance of 2(L/2) = 200mm = 0.2m at the speed of light in the free space c = 3e+8 m/s. The total travel time is therefore Δt = (0.2m)/(3e+8) = 6.67e-10 sec. Also, note that since the magnetic field at the probe location does not have any Z-component, the graph simply shows numerical noise.

To change the excitation waveform, you need to open the source's property dialog and click the Excitation Waveform button. The Excitation Waveform dialog give three options:

  • Automatically Generate Optimal Waveform
  • Use Custom Frequency Domain Specifications
  • Use Custom Time Domain Specifications

The first option above is the default options and represents a modulated Gaussian waveform, whose Fourier transform matches your specified center frequency and bandwidth. Select the second option: Use Custom Frequency Domain Specifications. From the Waveform Type drop-down list, select the Sinusoid option and accept all of its default parameters. Since a sinusoidal waveform never decays, it will cause the FDTD time marching loop to go on forever. Open the FDTD Engine Settings dialog and in its Termination Criterion section, select the second option Specify End Time. Set the number of time steps to 2000. Now run an FDTD simulation of your structure and graph the time domain E-field plot of your probe in EM.Grid.

The FDTD Excitation Waveform dialog showing a modulated Gaussian waveform.
The FDTD Excitation Waveform dialog showing a sinusoidal waveform.
Changing the convergence criterion in the FDTD Engine Settings dialog.
Cartesian graph of the Ez field component as a function of time with a sinusoidal waveform excitation.
Defining a Diode device in the Lumped Load dialog.

Analyzing a Diode

In this part of this tutorial lesson, you will place a nonlinear diode at the end of your transmission line. Open the property dialog of your lumped load and change its type to Diode. Accept the default parameter values of 100fA for Saturation Current and the room temperature of 300°K. Make sure the diode is defined to have +Z direction.

Next, move your Field Probe from the middle of the transmission line to its terminated end and place it at (100mm, 0, 1mm), that is exactly on top of your lumped load. Keep the sinusoidal waveform of the last part and run an FDTD analysis with a fixed number of 2000 time steps. Plot the E-field graph of the probe. You will notice a slight rounding at the maximum peaks of the output waveform. Now, go back to the Excitation Waveform dialog of your distributed source and change the Amplitude of your source from the default 1V to 10V. Run a new FDTD analysis and plot the E-field graph of the probe once again. This time you will find that the field waveform has been clipped from the top. This happens when the diode is reverse-biased. Keep in mind that the Ez field at the diode location is proportional to the voltage across the diode:

[math] V_d = \int{ E_z dz } \approx E_ z d [/math]

At negative cycles of the waveform, the positive peak is clapped at 360V/m. With d = 2mm, this yields Vd = (360V/m)(0.002m) = 0.72V, as you would expect from a typical diode.

Cartesian graph of the Ez field component at the diode location with a source amplitude of 1V.
Cartesian graph of the Ez field component at the diode location with a source amplitude of 10V.
EM.Cube's Functions dialog.

Investigating a Capacitive Load

In the last part of this tutorial lesson, you will place a 10pF capacitor at the termination load of your transmission line circuit. Open the property dialog of your lumped load and select Capacitor as the Load Type. Set the capacitance to 10pF. In this part, you will define a rectangular pulse waveform (or square wave) to excite your source. EM.Cube provides an extensive library of mathematical functions that you can use in a variety of application including waveform definition. You can see a list of these functions by clicking the Functions button of the Simulate Toolbar or using the keyboard shortcut Ctrl+I. Scroll down to locate the function Sqwv(x). This is a standard square wave function of unit period 1 oscillating between +1 and -1. Note that the function names are case-insensitive.

Next, open the Excitation Waveform dialog of your distributed source. Select the third radio button labeled Use Custom Time Domain Specifications and choose the Custom option of the Waveform Type drop-down list. Highlight the first row (Expression) of the waveform table. In the Expression box, enter the text sqwv(t/1n) and click the Accept button of the dialog. This will produce a square wave function of period 1ns as you can see from the graph on the right panel of the dialog.

Defining a capacitor in the Lumped Load dialog.
Defining a square wave temporal function in the Excitation Waveform dialog.

Before running an FDTD simulation, open the FDTD Engine Settings dialog and set the number of time steps to 4000. Then, run the FDTD analysis. Once the simulation is finished, open the Data Manager, locate the data file "DS_1_Source.DAT" and plot it in EM.Grid. This is the input voltage source waveform as you already saw in the Excitation Waveform dialog. Also plot the data file "Probe_1_E_Time.DAT". This graph represents the output voltage waveform across the capacitor.

Cartesian graph of the input voltage waveform at the source.
Cartesian graph of the Ez field component at the capacitor location, which is proportional to the output voltage waveform.




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